Contains report on one research project.National Institutes of Health (Grant 5 P01 GM14940-03

Johnson, D. H.

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XV. COMMUNICATIONS BIOPHYSICSAcademic and Research StaffL. D. BraidaS. K. BurnsH. S. ColburnL. S. FrishkopfJ. L. GoldsteinjtJ. J. Guinan, Jr.R. G. MarkSW. T. PeakeftBaerL. BennE. BerlinerCintronDemko, Jr.O. FrostGaimanHasanProf. W. M. SiebertProf. T. F. Weisst *,Dr. J. S. Barlowf~fMr. N. I. DurlachDr. R. D. Hallt - Dr. N. Y. S. Kiang?Dr. E. P. LindholmProf.Prof.Prof.Prof.Prof.Prof.Prof.Prof.T.D.J.R.P.D.B.Z.D. W. AltmanntR. M. BrownTF. H. ByerstD. J. CallahanA. H. CristTW. F. KelleyL. H. SeifelS. N. TandonC. MoxonNedzelnitskyM. RabinowitzV. ReedJ. ShillmanO. Stahl IIP. Tripp, Jr.R. WolfeA. STATISTICAL RELATIONSHIPS BETWEEN THE FIRINGOF TWO AUDITORY-NERVE FIBERSPATTERNSThe firing patterns of action potentials (spikes) in single auditory nerve fibers of the1-4cat have been studied extensively in our laboratory for the past few years. A knowl-edge of the relationships between firing patterns in different fibers, as well as the timingof spikes in each auditory-nerve fiber, is important for understanding the coding ofacoustic stimuli into spike trains. In this report we present results that suggest thatthe firing patterns of auditory-nerve fibers can be considered as independent randomprocesses. 5Data were obtained simultaneously from two auditory nerve fibers by meansof two micropipettes. The data were then analyzed "off-line" on a PDP-4 com-puter.This work was supported principally by the National Institutes of Health(Grant 5 P01 GM14940-03).tAlso at the Eaton-Peabody Laboratory of Auditory Physiology, Massachusetts Eyeand Ear Infirmary, Boston, Massachusetts.tInstructor in Medicine, Harvard Medical School, Boston, Massachusetts.Instructor in Preventive Medicine, Harvard Medical School, Boston, Massachusetts.ttResearch Affiliate in Communication Sciences from the Neurophysiological Labora-tory of the Neurology Service of the Massachusetts General Hospital, Boston,Massachusetts.QPR No. 97Graduate StudentsL. HicksJ. M. HoutsmaW. JamesH. JohnsonW. Kress, Jr.K. LewisY-S. Li173(XV. COMMUNICATIONS BIOPHYSICS)1. Analysis of Spike Activity Modeled as a Renewal Process: SpontaneousActivity and the Responses to High-Frequency TonesIn the absence of acoustic stimulation, a spontaneous sequence of action potentialscan be recorded from an auditory-nerve fiber. Statistical analyses show that the spon-taneous activity of auditory-nerve fibers can be modeled as a renewal process1,6 (i. e.the interspike intervals are modeled as independent, identically distributed, randomvariables). In response to high-frequency tonal stimuli (greater than 3-7 kHz), theinstantaneous rate of firing is also constant. Preliminary results 7 show that the spiketrains of auditory-nerve fibers in response to high-frequency tones can also be modeledas a renewal process.First, we shall explain our test for the independence of two renewal processes. Letus define T to be an instant of time chosen at random.DEFINITION. The interval u measured from time T until the next event in therenewal process occurs is called the forward recurrence time.The probability density function (pdf) of u, fu(L), is given byf (L) = f (x) dx, (1)u E[T] Lwhere f (x) is the probability density function of the interspike interval T, and E(T) isthe expected value of T.One way of estimating f (L) is to determine the interval distribution of the renewalprocess f (x), and then apply Eq. 1.A second method of estimating fu(L) makes direct use of the definition of the forwardrecurrence time. We can use a train of sampling pulses to define a sequence of times{Ti} from which we measure the forward recurrence times. If the sampling train isitself a renewal process independent of the sampled renewal process, this estimate willconverge to fu(L) as the number of samples becomes large. 8Our test for the independence of two renewal processes compares these two methodsof estimating fu(L). If the difference between the two estimates E(L) lies within some"allowable error" bound (AEB), we shall call the renewal processes statisticallyindependent. If E(L) crosses the AEB, we shall say that the renewal processes arenot independent. For the purposes of this analysis, we have chosen the AEB to betwo and one-half standard deviations of the error E(L). It can be shown 9 that thevariance of E(L) is of the formN L+AL +ALvar [E(L)] = 9 fu1- f (x) dx , (2)QPR No. 97 174ms 120(a)Zz0500-50Fig. XV-1. Analysis of the independence of the spontaneous activityof fiber pair K547-Al, B2. Number of spikes of unit B2(the sampling train) used in the analysis: 6997.(a) The estimate of f (L) for unit K547-Al using Eq. 1is shown as a heavy smooth line, while the estimateof fu(L) made with the second method is shown as ajagged line.(b) The AEB is shown as an envelope about E(L), the dif-ference between the two estimates of f (L).QPR No. 97ms(b)175(XV. COMMUNICATIONS BIOPHYSICS)where AL is the resolution of histograms used to estimate f (x), and N is the numberof samples in the estimate of f u(x).A typical result of the application of this test to auditory-nerve fiber data is shownin Fig. XV-1. For almost every value of L the estimate of the difference E(L) iswithin the allowable error bound (AEB). Hence, this pair of spike trains can be con-sidered statistically independent. Spontaneous activity was analyzed for a total of23 fiber pairs. The spontaneous rates ranged from 10 spikes/s to 105 spikes/s. Ofthis population, all fiber pairs but one were judged to be statistically independent.Eight fiber pairs have been considered in the analysis of responses to high-frequencytones. Seventeen stimulus presentations covering stimulus frequencies from 8 kHzto 25 kHz at sound levels ranging over 60 dB were made. All but one fiber pairwere judged to be statistically independent on the basis of our criterion.2. Analysis of Responses to Low-Frequency TonesFor low-frequency sinusoidal stimuli (less than 3-7 kHz) the instantaneous rate offiring of auditory-nerve fibers is periodic, with a period equal to the period of the stim-ulus. A stationary characterization of this periodic firing pattern can be obtained bydefining statistics that average over a period of the process. We need to calculatestatistics that depend upon the following functions:1. p(n), the probability of n spikes occurring in one period2. c(k), the correlation function of the number of events in period i and in periodi + k.The functions p(n) and c(k) were each estimated for 12 auditory-nerve fibers. Theexamples shown in Fig. XV-2 are typical and no fiber that was examined showedresults that deviated significantly from those shown. The results may be summarizedas follows.1. The probability of more than one spike in one period of the stimulus is much lessthan the probability of one spike.2. The correlation function reveals that the firings in intervals separated by morethan 4 periods of the stimulus are uncorrelated. Intervals separated by less than4 periods of the stimulus are negatively correlated.In order to test for the independence of two processes having the characteristicsdescribed above, we would need to know p(n) and c(k) for each process. To simplifythe calculations, we have made the following assumptions: (a) no more than one spikeoccurs in any one period of the stimulus, and (b) the firings in disjoint intervalsare independent. These assumptions allow us to describe the time-locked firing patternof an auditory-nerve fiber as a Bernoulli process. It can be shown that the AEB foundfrom the Bernoulli-process model is a lower bound on the AEB computed from the moreexact model with the p(n) and c(k) functions 9 used.QPR No. 97 176p(n)0.50 1 2 3c(k) (a)0.10.010.0010 1 2 3(b)Fig. XV-2. (a) The function p(n) is shown for unit K545-B17. Stimulusfrequency: 0. 499 kHz at a level of -50 dB re 200 V peak-to-peak to the earphone; number of periods of the stim-ulus used in this estimate: 28, 775.(b) The function c(k) is shown for unit K545-A12. Since c(k)is an even function, only the values for positive k areshown. Stimulus frequency: 1. 212 kHz at a level of-40 dB; number of periods of the stimulus used in thisestimate: 106, 280. A total of 7270 spikes occurred in therun.QPR No. 97 177SPIKES2000 J2000 -I I( b)2000-3 -2 -1 0 1 2 3(c)Fig. XV-3. Analysis of the independence of the time-locked activityof fiber pair K546-A8, B5. Number of joint occurrencesin periods of the two spike trains as a function of k isindicated by the heavy line. The expected number ofjoint occurrences (independence assumed) is indicatedby the light line. The dashed line denotes the AEB.Stimulus frequency: 0. 393 kHz. (a) Stimulus level =-60 dB; N = 34, 767. (b) Stimulus level = -50 dB; N =38, 183. (c) Stimulus level = -40. dB; N = 34, 816.QPR No. 97, , i I' ' " I I I I - k178(XV. COMMUNICATIONS BIOPHYSICS)The test for the independence of two Bernoulli processes (A and B) is based uponthe following property of joint probabilities: The probability of the joint occur-rence of two events is the product of the probabilities of each event if the eventsare independent. Therefore, if the Bernoulli processes are independent, theprobability of an event in trial i of process A and an event in trial (i-k) of pro-cess B is the product of the probability of an event in process A and the probabil-ity of an event in process B. Our test of independence is to measure the numberof events in each spike train and the number of joint events in intervals separatedby k periods in the two trains. The number of joint events can be estimated byassuming that the processes are independent. We have chosen the AEB placed uponthe excursions of the difference between the measured and estimated values of thenumber of joint events to be two and one-half standard deviations:AEB(k) = 2. 5 '/NPAPB(1-PA)(1-pB),where PA(PB ) is the probability of an event in spike train A(B) during one period ofthe stimulus, and N is the number of periods of the stimulus.Thirty-two fiber pairs were analyzed with a total of 119 stimulus presentationscovering frequencies from 0. 150 kHz to 7. 076 kHz over a sound level range of80 dB. A typical result of this analysis is shown in Fig. XV-3. All but three fiberpairs showed results consistent with statistical independence.3. ConclusionsFor our statistical tests, we would expect data from two independent processesto sometimes exceed the AEB. The population of fiber pairs that we examinedshowed a small number of cases in which the AEB was exceeded at one or twopoints. Thus we may conclude: This small number of fiber pairs represents asmall subpopulation (less than 10% of the total number) that has dependent firingpatterns, and/or these cases represent the occasional, expected crossings of theAEB. If the former were the explanation we might expect the statistical depen-dence of firing patterns of fiber pairs to be maintained for small changes in stim-ulus level. This we did not find for more than one case. Also the physiologicalcharacteristics (CF, spontaneous rate, threshold) of the fiber pairs that were foundto be dependent did not show any consistency. We, therefore, tentatively concludethat firing patterns of auditory-nerve fibers can be modeled as independent randomprocesses.Some interpretations of these results, as well as details of the statisticalanalyses, may be found in the author's thesis. 9D. H. JohnsonQPR No. 97 179(XV. COMMUNICATIONS BIOPHYSICS)References1. N. Y. S. Kiang, T. Watanabe, E. C. Thomas, and L. F. Clark, Discharge Pat-terns of Single Fibers in the Cat's Auditory Nerve (The M. I. T. Press, Cambridge,Mass., 1965).2. J. L. Goldstein and N. Y. S. Kiang, "Neural Correlates of the Aural Combina-tion Tone 2fl -f2," Proc. IEEE 56, 981-992 (1968).3. P. R. Gray, "A Statistical Analysis of Electrophysiological Data from AuditoryNerve Fibers in Cat," Technical Report 451, Research Laboratory of Elec-tronics, M. I. T., Cambridge, Mass., June 21, 1966.4. M. B. Sachs and N. Y. S. Kiang, "Two-Tone Inhibition in Auditory Nerve Fibers,"J. Acoust. Soc. Am. 43, 1120-1128 (1968).5. The data used in this analysis were obtained by Dr. N. Y. S. Kiang and thestaff of the Eaton-Peabody Laboratory, Massachusetts Eye and Ear Infirmary,Boston, Mass.6. D. R. Cox, Renewal Theory (John Wiley and Sons, Inc., New York, 1962).7. T. F. Weiss, Personal communication, 1969.8. S. C. Port and C. J. Stone, "A Large Sample Test for the Independence of TwoRenewal Processes," Ann. Math. Statist. 38, 1037-1041 (1967).9. D. H. Johnson, "Statistical Relationships between Firing Patterns of Two AuditoryNerve Fibers, " S. M. Thesis, Department of Electrical Engineering, M. I. T.,Cambridge, Mass., 1970.QPR No. 97 180180

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